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<h1>ADAP chromatogram deconvolution</h1>

<h2>Wavelets (ADAP)</h2>

<p>
    ADAP detects EIC peaks by using a CWT. Wavelet
    coefficients are first calculated as the inner product between the EIC and wavelets at different
    scales and locations. Subsequently, peak location and boundaries are
    determined through a ridgeline
    detection and simple local minima search. Though the algorithm uses a CWT is differs significantly from centWave
    primarily in the ridgline detection algorithm.
     <!-- Tautenhahn et al. [<a href="#ref1">1</a>]. -->
</p>

<h4>Ridgeline detection</h4>
A real peak in an EIC should create a local maxima in the wavelet coefficients at multiple scales.
The wavelet scale for which the wavelet most closely matches the shape of the peak, the best scale,
will create the
largest coefficient. Scales close to the best scale should also have reasonably similar shapes to
the peak and therefore create adjacent maxima between those scales.
Ridgelines are the series of connected local maxima across scales which are
indicative of a real peak.

Our procedure for detecting the ridgelines is similar to that described by Du et
al.[<a href="#ref1">1</a>] and Wee et al.[<a href="#ref2">2</a>] and is as follows:

<ol>
    <li>Begin with the coefficients corresponding to the largest wavelet scale.</li>

    <li> Find the largest coefficient at this scale and initialize a ridgeline.</li>

    <li> Remove all coefficients that are
        within half the estimated compact support of the Ricker wavelet (2.5 times
        the current scale).</li>

    <li> Find the next largest coefficient discounting all removed coefficients and initialize another ridgeline.</li>

    <li> Repeat steps (3)-(4) until there are no more coefficients remaining for this wavelet scale.</li>

    <li> Move to the next scale (decrease by one) and repeat (1)-(6). Add new coefficients to an existing
    ridgeline if they are close in RT. We define "close" to be a difference in their
    indices that is less than or equal to the current scale being investigated.</li>

    <li> After all scales have been processed, ridgelines must have a length, i.e., the total number of
    scales represented in the ridgeline, greater than or equal to 7.</li>
</ol>

<h4>Method parameters</h4>

<dl>
    <dt>S/N Threshold</dt>
    <dd>To calculate the signal (S) to noise (N) ratio,
        S/N, we choose S to be the maximum intensity between the boundaries of the peak under investigation.
        Noise, N, is estimated using two different steps. The final estimate of N is the
        smaller value found from the two steps and is used to calculate S/N. Each estimation of the
        noise attempts to avoid overestimate from the accidental inclusion of other real peaks that may be
        close in RT.

        Step 1:

        <ol>
        <li> Set two windows, one on each side of the peak in the EIC. The windows begin at
        the left and right peak boundaries and end at the peak boundaries plus or minus 2 times PW, respectively, where
        PW stands for peak width and is defined to be the number of scans between the two boundaries of
        a peak.</li>

            <li> Calculate the standard deviation of the intensities in the two combined windows and store it as
        one possible value of the noise.</li>

            <li> Expand both windows out from the peak by a single scan. The boundaries closest to the peak remain
        the same. After the first expansion, each window has a length of 2 times PW+1.</li>

            <li> Calculate and store the standard deviation of the intensities in the combined windows.</li>

            <li> Repeat steps (3)-(4) until each window has a length of 8 times PW.</li>

            <li> Incrementally shrink each window by one scan, calculating and storing the standard
        deviations of the combined windows. The windows are shrunk by moving the boundary closest
        to the peak toward the boundary furthest from it.</li>

            <li> Repeat step (6) until the window size is 2 times PW.</li>

            <li> The final noise estimate is taken to be the smallest stored standard deviation.</li>
        </ol>

        Step 2:

        <ol>
            <li> Same as (1) in step 1.</li>

            <li> Same as (2) in step 1.</li>

            <li> Shift each entire window away from the feature by one scan; the window lengths do not change.</li>

            <li> Repeat steps (2)-(3) until each window's boundary furthest from the feature is 8 times PW
            from the closest boundary of that feature.</li>

            <li> The final noise is taken to be the smallest stored standard deviation.</li>
        </ol>
    </dd>

    <dt>Min feature height</dt>
    <dd>The smallest intensity a peak can have and be considered a real
        feature.
    </dd>

    <dt>Coefficient/area threshold</dt>
    <dd>The best coefficient (largest inner product of wavelet with peak in ridgeline) divided by the area under the curve of the feature</dd>

    <dt>Peak duration range</dt>
    <dd>The acceptable range of peak widths. Peaks with widths outside this range will be rejected.</dd>

    <dt>RT wavelet range</dt>
    <dd>The range of wavelet scales used to build matrix of coefficients. Scales are expressed as RT values (minutes) and correspond to the range
        of wavelet scales that will be applied to the chromatogram. Choose a range that that simmilar to the range of peak widths expected to be found
        from the data.
    </dd>


</dl>

<h2>References</h2>

<p>
    <a name="ref1"></a>
    [1] Du, P., Kibbe W. A., and Lin S. M.,  <span style="font-style: italic;">Bioinformatics</span> <span style="font-weight: bold;">2006</span>,
    22:2059-65.
</p>

<p>
    <a name="ref2"></a>
    [2] Wee A., Grayden D. B., Zhu Y., Petkovic-Duran K., and Smith D.,  <span style="font-style: italic;">Electrophoresis</span>
    <span style="font-weight: bold;">2008</span>, 29:4215-25.
</p>


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